How do I make glowing line?
Animating the front of the curve
Assuming, as you say, that you have already created your Bezier Curve, all you need to do is:

make sure that the Curve has a Fill Type = Full and a Bevel Depth greater than zero

animate the Bevel End value using keyframes, from 0 at the start frame to 1 at the end frame.



Shading
You say that you have already found a shading solution. Let me share my solution anyway:

Enable &quot;Use UV for mapping&quot; in the Curve properties.

Use this node setup (the &quot;Add&quot; node is from &quot;Converter &gt; Vector Math&quot;).


Animate the Y coordinate of the Combine XYZ node. This will allow you to create the moving blobs of light along the line.